A square matrix $\mathbf{A}$ (of size $a\times a$) is not full rank and I want to decompose it as $\mathbf{A}=\mathbf{B}\mathbf{B}^\top$ where $\mathbf{B}$ is an $a\times(a-1)$ matrix of the $a-1$ eigenvectors of unit length corresponding to the nonzero eigenvalues of $\mathbf{A}$.
For example, $\mathbf{A}=\begin{pmatrix} 1/2 & -1/2\\ -1/2 & 1/2 \end{pmatrix}$ and $\mathbf{B}=(\sqrt{2}/2,-\sqrt{2}/2)^\top$ when $a=2$.
Question: Is there a name for this kind of matrix decomposition?